We construct time dependent BPS pp-wave brane solutions in the context of M-theory and type II supergravity. It is found that the N-brane solutions we consider satisfy the crossing rule as S-brane solutions but 1/8 supersymmetry remains. By applying them to the cosmological setting, inflationary solutions are obtained. During this inflation, the size of the extradimensions becomes smaller than our four-dimensional spacetime dynamically. We also discuss the mechanism for terminating this inflation and recovering the hot big-bang universe.
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